Career Summary

Qualifications

Research

Research keywords

Banach space geometry

Functional Analysis

Metric fixed point theory

Nonlinear Analysis

Ultraproducts of metric and Banach space

Research expertise

Sims' broad area of research is functional and nonlinear analysis, in which he has published over 60 refereed research papers in major international journals. More specifically his interest and expertise is in geometry in both Banach spaces and convex metric spaces, attainment problems and the use of embedding techniques (for example, embedding into ultrapowers) for solving such problems. His investigations have included attainment problems associated with boundary points of operator numerical ranges, and more recently the attainment of fixed points by various classes of mappings.

Nonlinear optimization and control leads to variational inequalities and hence to fixed points. Equilibria of dynamical systems correspond to fixed points, whose approximability relate to the stability and ergodic behaviour of the system. The research aims to further our understanding of nonexpansive type mappings, with an emphasis on establishing easily verified, yet widely applicable, criteria for the existence of a fixed together with effective iterative algorithms for their approximation. Of particular interest are the little understood cases where the underlying space admits no natural linear structure or nice geometry, so the usual tools of convex analysis are inapplicable. Geodesic metric spaces and notions of hyperbolicity are a fertile setting for investigations while ultraproduct methods provide powerful techniques.

Administrative

Administrative expertise

Teaching

Teaching keywords

All areas at undergraduate level

At all levels

Functional analysis at graduate level

Teaching expertise

Sims has taught undergraduate courses in most areas of pure mathematics, and some in applied mathematics; in particular mathematical modelling in the life sciences, operations research and optimization. These have been at all levels; pre-undergraduate to honours and to students of all ability levels.

Falset JG, Fuster EL, Sims B, 'Proceedings of the International Conference on Fixed Point Theory and Applications', Proceedings of the International Conference on Fixed Point Theory and Applications, Valencia, Spain (2004) [E4]